The term " Bayesian " refers to the 18th century mathematician and theologian Thomas Bayes, who provided the first mathematical treatment of a non-trivial problem of Bayesian inference.

The term Bayesian refers to Thomas Bayes ( 1702 – 1761 ), who proved a special case of what is now called Bayes ' theorem in a paper titled " An Essay towards solving a Problem in the Doctrine of Chances ".

Bayesian also refers to the application of this probability theory to the functioning of the brain

The term Bayesian refers to Thomas Bayes ( 1702 – 1761 ), who proved a special case of what is now called Bayes ' theorem.

The term MMSE specifically refers to estimation in a Bayesian setting, since in the alternative frequentist setting there does not exist a single estimator having minimal MSE.

Many modern machine learning methods are based on objectivist Bayesian principles.

In general, Bayesian methods are characterized by the following concepts and procedures:

In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of Markov chain Monte Carlo methods, which removed many of the computational problems, and an increasing interest in nonstandard, complex applications.

Nonetheless, Bayesian methods are widely accepted and used, such as in the fields of machine learning and talent analytics.

To meet the needs of science and of human limitations, Bayesian statisticians have developed " objective " methods for specifying prior probabilities.

Each of these methods has been useful in Bayesian practice.

Indeed, methods for constructing " objective " ( alternatively, " default " or " ignorance ") priors have been developed by avowed subjective ( or " personal ") Bayesians like James Berger ( Duke University ) and José-Miguel Bernardo ( Universitat de València ), simply because such priors are needed for Bayesian practice, particularly in science.

Thus, the Bayesian statistician needs either to use informed priors ( using relevant expertise or previous data ) or to choose among the competing methods for constructing " objective " priors.

In a clinical trial it is strictly not valid to conduct an unplanned interim analysis of the data by frequentist methods, whereas this is permissible by Bayesian methods.

Similarly, if funding is withdrawn part way through an experiment, and the analyst must work with incomplete data, this is a possible source of bias for classical methods but not for Bayesian methods, which do not depend on the intended design of the experiment.

Furthermore, as mentioned above, frequentist analysis is open to unscrupulous manipulation if the experimenter is allowed to choose the stopping point, whereas Bayesian methods are immune to such manipulation.

Bayesian methods would suggest that one hypothesis was more probable than the other, but individual Bayesians might differ about which was the more probable and by how much, by virtue of having used different priors ; but that's the same thing as disagreeing on significance levels, except significance levels are just an ad hoc device which are not really a probability, while priors are not only justified by the rules of probability, but there is definitely a normative methodology to define beliefs ; so even if a Bayesian wanted to express complete ignorance ( as a frequentist claims to do but does it wrong ), they could do it with the maximum entropy principle.

However, certain phenetic methods, such as neighbor-joining, have found their way into cladistics, as a reasonable approximation of phylogeny when more advanced methods ( such as Bayesian inference ) are too computationally expensive.

Recent research has shown that Bayesian methods that involve a Poisson likelihood function and an appropriate prior ( e. g., a smoothing prior leading to total variation regularization or a Laplacian prior leading to-based regularization in a wavelet or other domain ) may yield superior performance to expectation-maximization-based methods which involve a Poisson likelihood function but do not involve such a prior.

Using a parsimony criterion is only one of several methods to infer a phylogeny from molecular data ; maximum likelihood and Bayesian inference, which incorporate explicit models of sequence evolution, are non-Hennigian ways to evaluate sequence data.

In Bayesian statistics, however, the posterior predictive distribution can always be determined exactly — or at least, to an arbitrary level of precision, when numerical methods are used.

Several methods of Bayesian estimation select measurements of central tendency from the posterior distribution.

There is also an ever growing connection between Bayesian methods and simulation-based Monte Carlo techniques since complex models cannot be processed in closed form by a Bayesian analysis, while a graphical model structure may allow for efficient simulation algorithms like the Gibbs sampling and other Metropolis – Hastings algorithm schemes.

* Coherence ( philosophical gambling strategy ), analogous concept in Bayesian probability

Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities.

The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions whose truth or falsity is uncertain.

To evaluate the probability of a hypothesis, the Bayesian probabilist specifies some prior probability, which is then updated in the light of new, relevant data.

Bayesian probability interprets the concept of probability as " an abstract concept, a quantity that we assign theoretically, for the purpose of representing a state of knowledge, or that we calculate from previously assigned probabilities ," in contrast to interpreting it as a frequency or " propensity " of some phenomenon.

Nevertheless, it was the French mathematician Pierre-Simon Laplace, who pioneered and popularised what is now called Bayesian probability.

Broadly speaking, there are two views on Bayesian probability that interpret the probability concept in different ways.

In the Bayesian view, a probability is assigned to a hypothesis, whereas under the frequentist view, a hypothesis is typically tested without being assigned a probability.

In Bayesian statistics, a probability can be assigned to a hypothesis that can differ from 0 or 1 if the truth value is uncertain.

Broadly speaking, there are two views on Bayesian probability that interpret the ' probability ' concept in different ways.

For objectivists, probability objectively measures the plausibility of propositions, i. e. the probability of a proposition corresponds to a reasonable belief everyone ( even a " robot ") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by requirements of rationality and consistency.

The objective and subjective variants of Bayesian probability differ mainly in their interpretation and construction of the prior probability.

Early Bayesian inference, which used uniform priors following Laplace's principle of insufficient reason, was called " inverse probability " ( because it infers backwards from observations to parameters, or from effects to causes ).

In fact, there are non-Bayesian updating rules that also avoid Dutch books ( as discussed in the literature on " probability kinematics " following the publication of Richard C. Jeffrey's rule, which is itself regarded as Bayesian ).

Jaynes in the context of Bayesian probability

simple: Bayesian probability

Bayesian versus Frequentist interpretations of probability.

According to the objectivist view, the rules of Bayesian statistics can be justified by requirements of rationality and consistency and interpreted as an extension of logic.

Despite the growth of Bayesian research, most undergraduate teaching is still based on frequentist statistics.

* Conjugate prior, in Bayesian statistics, a family of probability distributions that contains a prior and the posterior distributions for a particular likelihood function ( particularly for one-parameter exponential families )

As with other branches of statistics, experimental design is pursued using both frequentist and Bayesian approaches: In evaluating statistical procedures like experimental designs, frequentist statistics studies the sampling distribution while Bayesian statistics updates a probability distribution on the parameter space.

Estimators that incorporate prior beliefs are advocated by those who favor Bayesian statistics over traditional, classical or " frequentist " approaches.

Bayesian statistics is inherently sequential and so there is no such distinction.

* Bayesian statistics

Statisticians of the opposing Bayesian school typically accept the existence and importance of physical probabilities, but also consider the calculation of evidential probabilities to be both valid and necessary in statistics.

Those who promote Bayesian inference view " frequentist statistics " as an approach to statistical inference that recognises only physical probabilities.

The most important distinction between the frequentist and Bayesian paradigms, is that frequentist makes strong distinctions between probability, statistics, and decision-making, whereas Bayesians unify decision-making, statistics and probability under a single philosophically and mathematically consistent framework, unlike the frequentist paradigm which has been proven to be inconsistent, especially for real-world situations where experiments ( or " random events ") can not be repeated more than once.

Bayesians would argue that this is right and proper — if the issue is such that reasonable people can put forward different, but plausible, priors and the data are such that the likelihood does not swamp the prior, then the issue is not resolved unambiguously at the present stage of knowledge and Bayesian statistics highlights this fact.

In the Bayesian interpretation, Bayes ' theorem is fundamental to Bayesian statistics, and has applications in fields including science, engineering, economics ( particularly microeconomics ), game theory, medicine and law.

In statistics, Bayesian inference is a method of inference in which Bayes ' rule is used to update the probability estimate for a hypothesis as additional evidence is learned.

Bayesian updating is an important technique throughout statistics, and especially in mathematical statistics: Exhibiting a Bayesian derivation for a statistical method automatically ensures that the method works as well as any competing method, for some cases.